Poorly, while peat would be adequate as a “casing” (a top level of light soil that holds in moisture yet has so little nutrient value) that being said without destroying the natural bog by pulling up the peat and using it as a top cover its not going to be useful to you.
On top of that, peat bogs are incredibly INCREDIBLY valuable to this Earth and humanity in general, please don't destroy it. These bogs pull carbon out of the atmosphere and can hold an absolute metric $#^$ ton of it which is super important for the environment especially in this day in age.
Well we know it has some nutritional value right? Otherwise, why would mushrooms be growing there naturally?, wrong! lol In reality what feeds these mushroom crops you see in the moss is usually dead animals, pollens from flowers or minerals being corroded from rocks and whatnot (or a mix of all these). What these bogs can do (as I am sure you know) is preserve organic materials for ages and ages. Therefore, when you see a mushroom outcrop in a peat bog does it really seem so unlikely that it could be feeding off of a fox that actually died 30 years ago and just it just so happens that this year a spore finally found it?
Peat moss is used heavily in some regions of the world but only as a "casing" because while it provides a great matrix it is highly acidic (meaning you will have to add chemicals to lower the pH) and it is great for retaining moisture. However, its composed of dead sphagnum moss which leaves it essentially neutral in terms of nutrients for plant life.